# How do I compare two squared (semi-partial) correlation coefficients (R-squareds) from different samples?

Zou (2007) presents two methods of computing a confidence interval for the *difference* in R-squareds obtained from regressions on different data sets. If zero is not contained in the confidence interval then the two R-squareds do not differ statistically at the (two-tailed) input alpha level. Zou suggests for samples under 100 that a modified version of the traditional delta method be used to compute confidence intervals. Both methods may be computed using a spreadsheet. Zou also gives formulae for obtaining confidence intervals and tests for comparing two correlations. Thom Baguley has R code (here) for obtaining the 95% confidence intervals for differences between overlapping and independent correlations using the suggestion of Zou (2007) based upon methods described in Chapter 6 of Baguley (2012).

At the above R program link Thom also adds pointers to some robust methods available from Rand Wilcox's web pages (some of which are essentially identical to Zou's approach but replacing the input with upper and lower bounds derived from bootstrap CIs rather than using Fisher's z).

There is also downloadable MS-DOS software and an on-line calculator to evaluate confidence intervals for a single R-squared and a spreadsheet based on formulae from Smithson. SAS and SPSS syntax from Zou to compute confidence intervals for a single R-squared is also available.

R-squared is the square of the semi-partial correlation and so the above tests are equivalent to comparing two semi-partial correlations. The proportion of the outcome variance which is accounted for by a particular predictor or set of predictors beyond that accounted for by other predictors.

References

Baguley, T. (2012). Serious stats: A guide to advanced statistics for the behavioral sciences. Basingstoke: Palgrave. R code for methods outlined in Chapter 6 of this book comparing correlations with and without variables in common is given here. This also includes the method of Zou(2007) for computing confidence intervals for differences in correlations.

Zou, GY (2007) Toward using confidence intervals to compare correlations. *Psychological methods* **12(4)** 399-413. (available via the Psycnet APA website for CBSU users).